Gerald W. Hohmann

Three-dimensional induced polarization and electromagnetic modeling

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response va...

Magnetotelluric responses of three-dimensional bodies in layered earths

The electromagnetic fields scattered by a three‐dimensional (3-D) inhomogeneity in the earth are affected strongly by boundary charges. Boundary charges cause normalized electric field magnitudes, and thus tensor magnetotelluric (MT) apparent resistivities, to remain anomalous as frequency approaches zero. However, these E‐field distortions below certain frequencies are essentially in‐phase with the incident electric field. Moreover, normalized secondary magnetic field amplitudes over a body ultimately decline in proportion to the plane‐wave impedance of the layered host. It follows that tipper element magnitudes and all MT function phases become minimally affected at low frequencies by an inhomogeneity. Resistivity structure in nature is a collection of inhomogeneities of various scales, and the small structures in this collection can have MT responses as strong locally as those of the large structures. Hence, any telluric distortion in overlying small‐scale extraneous structure can be superimposed to ar...

Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations

We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodie...

A finite-difference, time-domain solution for three-dimensional electromagnetic modeling

We have developed a finite-difference solution for three-dimensional (3-D) transient electromagnetic problems. The solution steps Maxwells equations in time using a staggered-grid technique. The time-stepping uses a modified version of the Du Fort-Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first-order Maxwells equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence-free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times.An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth-order difference method at early times and a second-order method at other times. Numerical checks against analytical, integral-equation, and spectral differential-difference solutions show that the solution provides accurate results.Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains 100 X 100 X 50 grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Omega . m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much-needed insight about TEM responses in complicated 3-D situations.

Two‐dimensional resistivity inversion

Resistivity data on a profile often must be interpreted in terms of a complex two‐dimensional (2-D) model. However, trial‐and‐error modeling for such a case can be very difficult and frustrating. To make interpretation easier and more objective, we have developed a nonlinear inversion technique that estimates the resistivities of cells in a 2-D model of predetermined geometry, based on dipole‐dipole resistivity data. Our numerical solution for the forward problem is based on the transmission‐surface analogy. The partial derivatives of apparent resistivity with respect to model resistivities are equal to a simple function of the currents excited in the transmission surface by transmitters placed at receiver and transmitter sites. Thus, for the dipole‐dipole array the inversion requires only one forward problem per iteration. We use the Box‐Kanemasu method to stabilize the parameter step at each iteration. We have tested our inversion technique on synthetic and field data. In both cases, convergence is rapi...

ELECTROMAGNETIC SCATTERING BY CONDUCTORS IN THE EARTH NEAR A LINE SOURCE OF CURRENT

A theoretical solution is developed for the electromagnetic response of a two‐dimensional inhomogeneity in a conductive half‐space, in the field of a line source of current. The solution is in the form of an integral equation, which is reduced to a matrix equation, and solved numerically for the electric field in the body. The electric and magnetic fields at the surface of the half‐space are found by integrating the half‐space Green’s functions over the scattering currents. One advantage of this particular numerical technique is that it is necessary to solve for scattering currents only in the conductor and not throughout the half‐space. The response of a thin, vertical conductor is studied in some detail. Because the only interpretational aids available previously were scale model results for conductors in free space, the results presented here should be useful in interpreting data and in designing new EM systems. As expected, anomalies decay rapidly as depth of burial is increased, due to attenuation in...

Integral equation modeling of three-dimensional magnetotelluric response

We have adapted a three‐dimensional (3-D) volume integral equation algorithm to magnetotelluric (MT) modeling. Incorporating an integro‐difference scheme increases accuracy somewhat. Utilizing the two symmetry planes of a buried prismatic body and a normally incident plane wave source greatly reduces required computation time and storage. Convergence checks and comparisons with one‐dimensional (1-D) and two‐dimensional (2-D) models indicate that our results are valid. We show theoretical surface anomalies due to a 3-D prismatic conductive body buried in a half‐space earth. Instead of studying the electric and magnetic fields, we have obtained impedance tensor and magnetic transfer functions by imposing two different source polarizations. Manipulation of the impedance tensor and magnetic transfer functions yields the following MT quantities: apparent resistivity and phase, impedance polar diagrams, tipper direction and magnitude, principal directions, skew, and ellipticity. With our preliminary analyses of...

Diffusion of electromagnetic fields into a two-dimensional earth; a finite-difference approach

We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space...

Transient electromagnetic inversion: A remedy for magnetotelluric static shifts

Surficial bodies can severely distort magnetotelluric (MT) apparent resistivity data to arbitrarily low frequencies. This distortion, known as the MT static shift, is due to an electric field generated from boundary charges on surficial inhomogeneities, and persists throughout the entire MT recording range. Static shifts are manifested in the data as vertical, parallel shifts of log‐log apparent resistivity sounding curves, the impedance phase being unaffected. Using a three‐dimensional (3-D) numerical modeling algorithm, simulated MT data with finite length electrode arrays are generated. Significant static shifts are produced in this simulation; however, for some geometries they are impossible to identify. Techniques such as spatial averaging and electromagnetic array profiling (EMAP) are effective in removing static shifts, but they are expensive, especially for correcting a previously collected MT data set. Parametric representation and use of a single invariant quantity, such as the impedance tensor ...

Topographic effects in resistivity and induced-polarization surveys

We have made a systematic study of dipole‐dipole apparent resistivity anomalies due to topography and of the effect of irregular terrain on induced‐polarization (IP) anomalies, using a two‐dimensional (2-D), finite‐element computer program. A valley produces a central apparent resistivity low in the resistivity pseudosection, flanked by zones of higher apparent resistivity. A ridge produces just the opposite anomaly pattern—a central high flanked by lows. A slope generates an apparent resistivity low at its base and a high at its top. Topographic effects are important for slope angles of 10 degrees or more and for slope lengths of one dipole‐length or greater. The IP response of a homogeneous earth is not affected by topography. However, irregular terrain does affect the observed IP response of a polarizable body due to variations in the distance between the electrodes and the body. These terrain‐induced anomalies can lead to erroneous interpretations unless topography is included in numerical modeling. A...